To determine the correct statement regarding the independence or dependence of events [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we need to understand the concept of conditional probability and independence.
1. Given Information:
- [tex]\( P(A) = 0.67 \)[/tex]: The probability that Edward purchases a video game.
- [tex]\( P(B) = 0.74 \)[/tex]: The probability that Greg purchases a video game.
- [tex]\( P(A \mid B) = 0.67 \)[/tex]: The probability that Edward purchases a video game given that Greg has purchased a video game.
2. Key Concept:
- Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent if and only if [tex]\( P(A \mid B) = P(A) \)[/tex].
3. Analysis:
- We are given that [tex]\( P(A \mid B) = 0.67 \)[/tex] and [tex]\( P(A) = 0.67 \)[/tex].
- Since [tex]\( P(A \mid B) = P(A) \)[/tex], this satisfies the condition for independence.
4. Conclusion:
- Therefore, [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent events because the condition [tex]\( P(A \mid B) = P(A) \)[/tex] holds true.
Given the provided information and the proper application of the concept of independence, the correct statement is:
A. Events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent because [tex]\( P(A \mid B) = P(A) \)[/tex].
